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The air entering the disused chimney at the bottom is warmer than the air within the chimney, therefore the RH of the air changes in the process.
Again from my paper, (with appropriate references as included in my original paper, woody!)
View attachment 142398
: http://ocw.usu.edu
A change of about 10°C doubles or halves the RH.
The warm entering at the bottom may be at about 20 - 25°C at about 45 -75% RH. It has the ability to absorb more moisture, especially from the fabric of the building ( the chimney) which may have a high moisture content.
The warm'ish air with a high RH is then ventilated out the top of the chimney, carrying with it any excess moisture that it has accumulated.
Warm air naturally rises.
BTW, in an unoccupied property the air at the bottom of the chimney will be no warmer than the air at the top or within the chimney. Therefore there will be no natural draught, or even a downward draught.